LIMIT THEOREMS FOR ONE AND TWO DIMENSIONAL RANDOM WALKS IN RANDOM SCENERY

LIMIT THEOREMS FOR ONE AND TWO-DIMENSIONAL RANDOM WALKS IN RANDOM SCENERY FABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, AND FRANÇOISE PÈNE Abstract. Random walks in random scenery are processes defined by Zn := ∑n k=1 ?X1+...+Xk , where (Xk, k ≥ 1) and (?y, y ? Zd) are two independent sequences of i.i.d. random variables with values in Zd and R respectively. We suppose that the distributions of X1 and ?0 belong to the normal basin of attraction of stable distribution of index ? ? (0, 2] and ? ? (0, 2]. When d = 1 and ? 6= 1, a functional limit theorem has been established in [16] and a local limit theorem in [7]. In this paper, we establish the convergence in distribution and a local limit theorem when ? = d (i.e. ? = d = 1 or ? = d = 2) and ? ? (0, 2]. Let us mention that functional limit theorems have been established in [3] and recently in [10] in the particular case when ? = 2 (respectively for ? = d = 2 and ? = d = 1). 1. Introduction Random walks in random scenery (RWRS) are simple models of processes in disordered media with long-range correlations.

• zn


• skorohod j1- topology


• when ?


• local limit


• random scenery


• characteristic function


• stable distribution


• limit theorems